On the Betti Numbers of some Classes of Binomial Edge Ideals

Abstract

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs depending on induced subgraphs.

Depth and extremal Betti number of binomial edge ideals

“…Another homological invariant that helps in understanding more about its structure is the Betti number. There have been few attempts in computing the Betti numbers of binomial edge ideals, for example, Zafar and Zahid for cycles, [24], Schenzel and Zafar for complete bipartite graphs, [23], Jayanthan et al. for trees and unicyclic graphs [12].…”

On the Extremal Betti Numbers of Binomial Edge Ideals of Block Graphs

“…Recently this has been confirmed to be true, if the PI graph consists of at most two cliques [2]. In general the graded Betti numbers are known only for very special classes of graphs including cycles [18].…”

Regularity of binomial edge ideals of certain block graphs

“…Then the ideal J G generated by {x i y j − x j y i | (i, j) is an edge in G} is called the binomial edge ideal of G. This was introduced by Herzog et al, [8] and independently by Ohtani, [12]. Recently, there have been many results relating the combinatorial data of graphs with the algebraic properties of the corresponding binomial edge ideals, see [1], [2], [4], [11], [14], [15], [17]. In particular, there have been active research connecting algebraic invariants of the binomial edge ideals such as Castelnuovo-Mumford regularity, depth, Betti numbers etc., with combinatorial invariants associated with graphs such as length of maximal induced path, number of maximal cliques, matching number.…”